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Proposition

Let $X$ be an absolute neighbourhood retract (ANR) and $A \xhookrightarrow{i} X$ a closed subspace-inclusion. Then $i$ is a Hurewicz cofibration iff $A$ is itself an ANR.

(Aguilar, Gitler & Prieto 2002, Thm. 4.2.15)

Proposition

Let $X$ be a paracompact Banach manifold. Then the inclusion $A \hookrightarrow X$ of any closed sub-Banach manifolds is a Hurewicz cofibration.

Proof

Being a closed subspace of a paracompact space, $A$ is itself paracompact (by this Prop.). But paracompact Banach manifolds are absolute neighbourhood retracts (this Prop.) Therefore the statement follows with Prop. .

$\pi_\ast$

Proposition

If $f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet$ is a morphism of simplicial spaces such that

1. on simplicial sets of connected components it is a Kan fibration;

2. all component spaces $X_n$, $Y_n$ ($n \in \mathbb{N}$) are connected or discrete topological space

then the geometric realization of any homotopy pullback-square of $f_\bullet$ is a homomotopy pullback?-square in topological spaces.

Write $TopSp$ for the convenient category of compactly generated weak Hausdorff spaces, and $TopSp_{Qu}$ for the classical model structure on topological spaces in its version for compactly generated spaces (this Thm.).

Proposition

If $H \subset G \,\in\, Grp(TopSp)$ is a subgroup-inclusion of topological groups such that the corresponding coset space coprojection is a Serre fibration

$$G \xrightarrow{ \;\in Fib\; } G/H \;\;\; \in \; TopSp_{Qu}$$

(for example in that it admits local sections).

Then the quotient of the universal principal space $E G$ by the subgroup $H$ is weak homotopy equivalent to the classifying space $B H$ of $H$:

$$(E G)/H \;\simeq\; B H \;\;\; \in Ho(TopSp_{Qu}) \,.$$

$$H \xrightarrow{\;} \frac{E G \times G}{G} \xrightarrow{\;} \frac{ E G \times (G/H) }{G}$$

Proposition

Transported through the equivalence of Prop. , the canonical group action (see this Prop.) of the Weyl group $W_G(H)$ on the $H$-fixed locus $Fnctr\big(\mathbf{E}G ,\, \mathbf{B}\Gamma\big)^H$ becomes, on connected components $\pi_0 \big( CrsHom(H,\,\Gamma) \sslash_{\!\!ad} \Gamma \big) \;\; \simeq \;\; H^1_{Grp}(H,\,\Gamma)$, the $W_G(H)$-action on the non-abelian group 1-cohomology of $H$ from Prop. .

Proof

We make explicit use of the functors $L, R$ constructed in the proof of Prop. . Noticing that $R$ is a section of $L$, we need to (1) send a crossed homomorphism up with $R$, (2) there act on it with $n$, (3) send the result back with $L$. The result is the desired induced action.

Explicitly, by the definition of $L$ in the proof of Prop. , this way a crossed homomorpism $\phi \,\colon\, H \to \Gamma$ is sent by $n \in N_G(H)$ to the assignment

(1)$$h \,\mapsto\, \Big(L\big( n \cdot (R \phi) \big)\Big)(h) \;=\; \alpha(n) \Big( (R\phi) \big( n^{-1} ,\, n^{-1} \cdot h \big) \Big) \,.$$

It just remains to evaluate the right hand side.

Notice that the definition of $L$ is independent of the choice of $\sigma \,\colon\, G/H \xrightarrow{\;} G$ (?), and that $R$ (whose definition does depend on this choice ) is a section for each choice. Hence we may choose $\sigma$ in a way convenient way for each $n$.

Now if $n \in H \subset N_G(H)$ then its canonical action on the $H$-fixed locus is trivial, and also the claimed induced action is trivial, so that in this case there is nothing further to be proven. Therefore we assume now that $n$ is not in $H$, and then we choose $\sigma$ such as to pick $n^{-1}$ as the representative in its $H$-coset:

$$\sigma\big( \big[n^{-1}\big] \big) \;\coloneqq\; n^{-1} \,.$$

With this choice, the right hand side of (1) is evaluated as follows, where we repeatedly use that, by definition and choice of $\sigma$, $R\phi$ assigns the neutral element to the morphism $n^{-1} \to \mathrm{e}$ in the pair groupoid:

$$\begin{aligned} \Big(L\big( n \cdot (R \phi) \big)\Big)(h) & \;=\; \alpha(n) \Big( (R\phi) \big( n^{-1} ,\, n^{-1} \cdot h \big) \Big) \\ & \;=\; \;=\; \alpha(n) \big( (R\phi)(\mathrm{e},\, n^{-1} \cdot h) \big) \\ & \;=\; \alpha\big(n^{-1} \cdot h \cdot n \cdot n\big) \big( (R\phi)(n^{-1} \cdot h^{-1} \cdot n,\, n^{-1}) \big) \\ & \;=\; \alpha\big(n^{-1} \cdot h \cdot n \cdot n\big) \big( (R\phi)(n^{-1} \cdot h^{-1} \cdot n,\, \mathrm{e}) \big) \\ & \;=\; \alpha(n) \big( (R\phi)(\mathrm{e},\, n^{-1} \cdot h \cdot n) \big) \\ & \;=\; \alpha(n) \big( \phi(n^{-1} \cdot h \cdot n) \big) \mathrlap{\,.} \end{aligned}$$

This is indeed the claimed formula (?).

Given $\alpha \,G\, \xrightarrow{\;} Aut_{Grp}(\Gamma)$ as above, consider a subgroup $H \subset G$.

Proposition

The set of crossed homomorphisms $H \to \Gamma$, with respect to the restricted action of $H$ on $\Gamma$, carries a group action of the normalizer subgroup $N_G(H) \,\subset\, G$, given by

$$\array{ N_G(H) \times CrsHom(H,\,G) &\xrightarrow{\;\;}& CrsHom(H,\,G) \\ (n, \phi) &\mapsto& \phi_{n} \mathrlap{ \;\coloneqq\; \alpha(n) \Big( \phi \big( n^{-1} \cdot (-) \cdot n \big) \Big) } }$$

Moreover, on crossed-conjugation classes of crossed homomorphisms, hence on first non-abelian group cohomology, this action descends to an action of the Weyl group $W_G(H) \coloneqq N_G(H)/H$:

$$H^1_{Grp}(H,\,\Gamma) \;\;\; \in \; W_G(H) Act(Sets) \,.$$

Proof

For the first statement: It is clear that this is a group action if only $\phi_n$ is indeed a crossed homomorphism. This follows by a direct computation:

$$\begin{array}{lll} \phi_n( h_1 \cdot h_2 ) & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot h_2 \cdot n \big) \Big) & \text{definition of}\; \phi_n \\ & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot n \cdot n^{-1} \cdot h_2 \cdot n \big) \Big) & \text{group property of}\; N_G(H) \subset G \\ & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot n \big) \cdot \alpha(n^{-1} \cdot h_1 \cdot n) \big( \phi( n^{-1} \cdot h_2 \cdot n ) \big) \Big) & \text{crossed homomorphism property of} \; \phi \\ & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot n \big) \Big) \cdot \alpha(h_1 \cdot n) \Big( \phi\big( n^{-1} \cdot h_2 \cdot n \big) \Big) & \text{action property of} \; \alpha \\ & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h_1 \cdot n \big) \Big) \cdot \alpha(h_1) \bigg( \alpha(n) \Big( \phi\big( n^{-1} \cdot h_2 \cdot n \big) \Big) \bigg) & \text{action property of} \; \alpha \\ & \;=\; \phi_n(h_1) \cdot \alpha(h_1) \big( \phi_n(h_2) \big) & \text{definition of}\; \phi_n \mathrlap{\,.} \end{array}$$

To see that this action descends to group cohomology, we need to show for

$$\phi'(-) \;=\; \gamma^{-1} \cdot \phi(-) \cdot \alpha(-)(\gamma)$$

a crossed conjugation, that there exists a crossed conjugation between $\phi'_n$ and $\phi_n$. The following direct computation shows that this is given by crossed conjugation with $\alpha(n)(\gamma)$:

$$\begin{array}{lll} \phi'_n(h) & \;=\; \alpha(n) \Big( \gamma^{-1} \cdot \phi\big( n^{-1} \cdot h \cdot n \big) \cdot \alpha \big( n^{-1} \cdot h \cdot n \big) (\gamma) \Big) & \text{assumption with definition of} \; \phi_n \\ & \;=\; \alpha(n) \big( \gamma^{-1} \big) \cdot \alpha(n) \Big( \phi\big( n^{-1} \cdot h \cdot n \big) \Big) \cdot \alpha(h) \Big( \alpha(n)(\gamma) \Big) & \text{action property of} \; \alpha \\ & \;=\; \big( \alpha(n)(\gamma) \big)^{-1} \cdot \phi_n(h) \cdot \alpha(h) \big( \phi(n)(\gamma) \big) & \text{definition of} \; \phi_n \,. \end{array}$$

To conclude, we need to show that for $n \in H \subset N(H)$ there is a crossed conjugation between $\phi_n$ and $\phi$. The following direct computation shows that this is given by crossed conjugation with $\phi(n)$ (which is indeed defined, by the assumption that $n \in H$):

$$\begin{array}{lll} \phi_n(h) & \;=\; \alpha(n) \Big( \phi \big( n^{-1} \cdot h \cdot n \big) \Big) & \text{definition of} \; \phi_n \\ & \;=\; \alpha(n) \bigg( \phi(n^{-1}) \cdot \alpha(n^{-1}) \Big( \phi(h) \cdot \alpha(h) \big( \phi(n) \big) \Big) \bigg) & \text{crossed homomorphism property of} \; \phi \\ & \;=\; \alpha(n) \big( \phi(n^{-1}) \big) \cdot \phi(h) \cdot \alpha(h) \big( \phi(n) \big) & \text{action property of} \; \alpha \\ & \;=\; \big( \phi(n) \big)^{-1} \cdot \phi(h) \cdot \alpha(h) \big( \phi(n) \big) & \text{crossed homomorphism property of} \; \phi \mathrlap{\,.} \end{array}$$

Let $\phi \;\colon\; G \xrightarrow{\;} \Gamma$ be a crossed homomorphism, hence equivalently a plain group homomorphism $(\phi(-),\,(-)) \,\colon\, G \xrightarrow{\;} \Gamma \rtimes G$. Say that another crossed homomorphism $\phi'$ is nearby if it is so as a plain homomorphism $(\phi'(-),(-))$.

Then the above theorem says that there is an element $(\gamma,\,h) \,\in\, \Gamma \rtimes G$ such that

$$\big( \phi'(g),\,g \big) \;\; = \;\; \big( \gamma, \, h \big)^{-1} \cdot \big( \phi'(g),\,g \big) \cdot \big( \gamma, \, h \big) \,.$$

In order for such a conjugation to be a crossed conjugation of the original morphism, we need $h = \mathrm{e}$.

Notice that we know already that $h \in C(G)$ is in the center of $G$, since the projection of both sides of the equation to $G$ must be the identity, by construction of crossed homomorphisms.

Hence the further conjugation of the above equation by $\big(\mathrm{e},\,h^{-1}\big)$ yields:

$$\Big( \alpha(h^{-1}) \big( \phi'(g),\, g \big) \Big) \;\; = \;\; \big( \mathrm{e},\, h \big) \cdot \big( \gamma, \, h \big)^{-1} \cdot \big( \phi'(g),\,g \big) \cdot \big( \gamma, \, h \big) \cdot \big( \mathrm{e},\, h^{-1} \big) \,.$$

Therefore, if the action of $G$ on $\Gamma$ restricts along the inclusion $C(G) \xhookrightarrow{\;} G$ to the trivial action, then

$$\big( \gamma, \, h \big) \cdot \big( \mathrm{e},\, h^{-1} \big) \;\; = \;\; \big( \gamma, \, \mathrm{e} \big)$$

corresponds to a crossed conjugation

$$\phi'(-) \;=\; \gamma^{-1} \cdot \phi(-) \cdot \alpha(-)(\gamma) \,.$$

crossed homomorphisms as sliced functors

Proposition

Internal to some ambient category $\mathcal{C}$ with finite limits, let

• $G \,\in\, Grp(\mathcal{C})$ be a group object,
• $P \,\in\, G Act(\mathcal{C})$ an action object,
• $(P \to X) \,\in\, G PsTor(\mathcal{C}_{/X})$ a formally principal bundle.

Then the following are equivalent:

1. $P \to X$ is the $G$-quotient coprojection;

2. $P \to X$ is an effective epimorphism.

---

• $G$-fixed-wise contractibility of $Maps(\mathbf{E}G,\mathbf{E}\Gamma)$ follows from $G$-equivariant contraction of $\mathbf{E}\Gamma$

• the universal equivariant principal $\infty$-bundle is $\ast \longrightarrow Maps(\mathbf{E}G, \mathbf{B}\Gamma)$ and the point is that the base space is pointed, but no longer pointed connected – but the universal bundle is still that point inclusion (meaning that all other fibers are empty, as admissible for a formally principal bundle)

---

Proof

The first condition is equivalent to

$$P \times_X P \rightrightarrows P \to X$$

being a coequalizer, the second to

$$P \times G \rightrightarrows P \to X$$

being a coequalizer. But the pseudo-principality condition says that we have an isomorphism (the shear map)

$$P \times_X P \simeq P \times G$$

which identifies these two diagrams.

$$Maps \big( \mathcal{X} ,\, \mathbf{E}G \big) \;=\; \Big( Fnctr \big( \mathcal{X} ,\, \mathbf{E}G \big) \times Fnctn \big( X_0 ,\, G \big) \rightrightarrows Fnctr \big( \mathcal{X} ,\, \mathbf{E}G \big) \Big)$$

$$Fnctr \big( \mathcal{X} ,\, \mathbf{E}G \big) \;\; \simeq \;\; Fnctr \big( \mathcal{X} ,\, \mathbf{B}G \big) \times G^{\pi_0(\mathcal{X})}$$